Why is the target $r + \gamma \max_{a'} Q(s', a'; \theta_i^-)$ in the loss function of the DQN architecture?
This is the problem that reinforcement learning (RL) is trying to solve: What is the best way to behave when we don’t know what the right action is and only have a scalar (the reward (r) is a scalar) reward of how well we have done?

RL approaches this problem by utilizing temporal difference learning and makes predictions based on the previous experience. An RL agent is trying to maximize the sum of future discounted rewards, called the return. 

The term $r + \gamma \max_{a'} Q(s', a'; \theta_i^-)$ is essentially saying "the reward I just saw + $\gamma$ * (my prediction of the return given that I take what I think is the best action in the current state and follow my policy from then on)". 

By updating the Q function, the agent can better predict the consequences of its actions and can then choose the best action with greater probability. 

The $\gamma$ (gamma) helps to balance between the immediate reward and future rewards. A $\gamma = 0$ makes the immediate reward the only thing that matters but usually the reward for a good action is delayed so values of gamma which put a higher importance on later rewards ($\gamma = .8, .9, .99$, etc) are used.